Arsenal's Mikel Arteta scored from just inside the box after Aaron Ramsey's intelligent cutback before Alex Song put the ball into his own net from Ruben Rochina's free-kick.
Yakubu tapped in Steven Nzonzi's shot across goal for 3-2, then Laurent Koscielny bundled past his own keeper after Martin Olsson's fine run and cross.
Marouane Chamakh's late headed consolation completed the scoring but Arsenal had late chances to rescue a point and had a last-gasp appeal for a penalty turned down.
The Rovers manager maintained throughout the week that he was the right man to turn around the club's fortunes, and this result will go some way to convincing the sceptics that he is up to the job.
His side were seemingly beaten at the break, being comprehensively outplayed and lucky to be only 2-1 down.
Woeful Blackburn defending allowed Song to find the totally unmarked Gervinho, whose decent low strike took a slight deflection on its way past Robinson in the home goal.
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But Rovers were posing their usual threat from set pieces, with Chris Samba twice heading wide when he should have hit the target while at the other end Robinson was keeping them in the game with saves from Andrey Arshavin and Gervinho.
Just as Arsenal seemed on the verge of adding to their lead, Blackburn grabbed a lifeline with debutant Yakubu scoring his first goal for the club with a deft finish past the on-rushing Wojciech Szczesny following a fine pass from David Hoilett.
Crucially though, Blackburn could not hold on until half-time. Arteta restored the visitors' lead, smashing the ball into the net from the edge of the box for his first Arsenal goal after Ramsey's pass.
Kean's men showed great spirit after the restart and totally outmuscled a fragile Arsenal unit in the second half.
They were back in the game soon after the break in adm